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How dielectric screening in two-dimensional crystals affects the convergence ofexcited-state calculations: Monolayer MoS2

Hüser, Falco; Olsen, Thomas; Thygesen, Kristian Sommer

Published in:Physical Review B Condensed Matter

Link to article, DOI:10.1103/PhysRevB.88.245309

Publication date:2013

Document VersionPublisher's PDF, also known as Version of record

Link back to DTU Orbit

Citation (APA):Hüser, F., Olsen, T., & Thygesen, K. S. (2013). How dielectric screening in two-dimensional crystals affects theconvergence of excited-state calculations: Monolayer MoS

2. Physical Review B Condensed Matter, 88(24),

[245309]. https://doi.org/10.1103/PhysRevB.88.245309

PHYSICAL REVIEW B 88, 245309 (2013)

How dielectric screening in two-dimensional crystals affects the convergenceof excited-state calculations: Monolayer MoS2

Falco Huser,1,* Thomas Olsen,1 and Kristian S. Thygesen1,2,†1Center for Atomic-scale Materials Design (CAMD), Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark

2Center for Nanostructured Graphene (CNG), Technical University of Denmark, 2800 Kgs. Lyngby, Denmark(Received 22 July 2013; revised manuscript received 29 October 2013; published 26 December 2013)

We present first-principles many-body calculations of the dielectric constant, quasiparticle band structure,and optical absorption spectrum of monolayer MoS2 using a supercell approach. As the separation between theperiodically repeated layers is increased, the dielectric function of the layer develops a strong q dependencearound q = 0. This implies that denser k-point grids are required to converge the band gap and exciton bindingenergies when large supercells are used. In the limit of infinite layer separation, here obtained using a truncatedCoulomb interaction, a 45 × 45 k-point grid is needed to converge the G0W0 band gap and exciton energy towithin 0.1 eV. We provide an extensive comparison with previous studies and explain agreement and variationsin the results. It is demonstrated that too coarse k-point sampling and the interactions between the repeatedlayers have opposite effects on the band gap and exciton energy, leading to a fortuitous error cancellation in thepreviously published results.

DOI: 10.1103/PhysRevB.88.245309 PACS number(s): 73.22.−f, 78.20.Bh, 71.20.Nr, 71.35.−y

I. INTRODUCTION

Atomically thin two-dimensional (2D) materials suchas graphene, hexagonal boron nitride, and transition-metaldichalcogenides (TMDC) possess unique electronic and op-tical properties including high intrinsic carrier mobilities,1–3

tunable band gaps,4,5 and strong light-matter interactions.6–9

These features, combined with the possibility of engineeringtheir electronic properties further via strain, alloying, orstacking, make the 2D materials ideal as building blocks fornew optoelectronic structures and devices with minimal sizesand performances surpassing present technologies.

After the intense focus on graphene, the TMDCs arenow attracting increasing interest.10 This stems mainly fromthe greater variation in their electronic properties includingboth semiconducting and metallic behavior. So far, the mostintensively studied single-layer TMDC is the semiconductorMoS2. Nanostructured forms of MoS2 have previously beenexplored as potential catalysts for desulferization of crudeoil and more recently for (photo-) electrochemical hydrogenevolution.11–13 Bulk MoS2 is composed of two-dimensionalsheets held together by weak van der Waals forces, andindividual sheets can be isolated by exfoliation techniquessimilar to those used to produce graphene.1 Single layers ofMoS2 therefore comprise highly interesting two-dimensionalsystems with a finite band gap and have recently been proposedfor nanoelectronics applications.2

The optical properties of bulk MoS2 have been thoroughlystudied experimentally.14–19 The absorption spectrum showstwo distinct low-energy peaks at 1.88 and 2.06 eV, which aredenoted by A and B, respectively,20 and derive from direct tran-sitions between a split valence band and the conduction bandat the K point of the Brillouin zone. Their Rydberg satellites,Zeeman splitting, and dependence on crystal thickness havebeen investigated in detail.17 Recently, the quantum yield ofluminescence from MoS2 was shown to increase dramaticallywhen the sample thickness was changed from a few layers to

a monolayer,7,8 indicating a transition to a direct band gap inthe single layer.

In the past couple of years a number of theoretical studiesof the electronic band structure and optical excitations inmonolayer MoS2 have been published.4,21–26 These studiesare based on many-body perturbation theory in the GW ap-proximation (mainly the non-self-consistent G0W0 approach)for the band structure and the Bethe-Salpeter equation (BSE)with a statically screened electron-hole interaction for theoptical excitations. As is standard practice, the calculationshave been performed on a supercell geometry where theMoS2 layers have been separated by 10–20 A vacuum andthe Brillouin zone (BZ) was sampled on grids ranging from6 × 6 to 15 × 15. With these parameters, G0W0 band gaps inthe range 2.6–3.0 eV and G0W0-BSE exciton binding energiesof 0.6–1.1 eV have been reported. Moreover, both direct21–25

and indirect4 band gaps have been found at the G0W0 level,while only direct gaps have been obtained with self-consistentGW 21 and GW0.4,26 When comparing these values, it shouldbe kept in mind that both the size and nature of the band gap ofMoS2 depend sensitively on the in-plane lattice parameter, a.4

One of the most fundamental quantities describing theelectronic structure of a material is the dielectric function. Thedielectric properties of atomically thin 2D materials are quitedifferent from their three-dimensional (3D) counterparts.27 Forexample, plasmons in 2D metals have acoustic dispersionrelations [ωp(q) → 0 as q → 0], and screening is generallymuch weaker, leading to strong exciton binding energies in2D semiconductors. Reported static dielectric constants formonolayer MoS2 obtained using the supercell approach liein the range 4.2–7.6 (for in-plane polarization).21,24,28 Thesevalues have been used to rationalize the exciton binding energyin MoS2 using the simple Mott-Wannier model.

In this paper, we present an in-depth study of the dielectricfunction, quasiparticle (QP) band structure, and excitonicstates in monolayer MoS2. We focus on separating the spuriousinterlayer screening from the intrinsic intralayer screening in

245309-11098-0121/2013/88(24)/245309(9) ©2013 American Physical Society

HUSER, OLSEN, AND THYGESEN PHYSICAL REVIEW B 88, 245309 (2013)

supercell geometries and on the consequences of the physics of2D screening for the convergence of many-body excited-statecalculations. The 3D macroscopic dielectric constant, as usedfor solids, converges to 1 for all q vectors in the limit ofinfinite separation of the layers and is thus meaningless for a2D material. We use an alternative approach to calculate thedielectric constant by averaging the total field over the materialrather than the supercell. This 2D dielectric constant showsstrong q dependence for small wave vectors and becomesexactly 1 for q = 0. This property has important consequencesfor the k-point convergence of many-body calculations.

In general, the use of a truncated Coulomb interaction isessential to avoid interlayer screening which decays slowlywith the layer separation, L. The interlayer screening yieldsa too large dielectric constant for wave vectors q < 1/L. Asa consequence, the G0W0 band gaps and exciton energies are0.5 eV too low on average for layer separations of around20 A. For larger layer separations, the strong q dependenceof the dielectric constant for small q implies that a k-pointgrid of at least 45 × 45 is required to converge band gaps andexciton energies to 0.1 eV. For k-point grids below 15 × 15the band gap is at least 0.5 eV too large in the limit L → ∞.Thus, the effects of interlayer screening and too coarse k-pointgrids partially cancel out, leading to reasonable values for theband gap and exciton binding energy with underconvergedparameters as applied in previous studies.

The paper is organized as follows. In Sec. II we presentG0W0 band structures and study the convergence of the gapwith respect to interlayer separation and k-point sampling. InSec. III we show calculations for the 2D dielectric constant andexplain the origin of the slow k-point convergence of the bandgap. In Sec. IV we present many-body calculations of the low-est excitons and analyze their convergence with layer separa-tion and k-point sampling. Our conclusions are given in Sec. V.

II. QUASIPARTICLE BAND STRUCTURE

In this section we demonstrate that GW band structuresfor monolayer MoS2 converge extremely slowly with re-spect to the interlayer separation. In order to obtain well-converged results (within 0.1 eV), the use of a truncatedCoulomb interaction is inevitable, along with a k-point grid ofaround 45 × 45. Previously reported calculations with the fullCoulomb interaction have employed only separation between10 and 20 A and used from 6 × 6 to 12 × 12 k points. Theresulting band structures are, however, somewhat saved by afortunate error cancellation between the two effects.

A. Computational details

All our calculations have been performed with the projectoraugmented wave method code GPAW.29,30 The Kohn-Shamwave functions and energies of monolayer MoS2 were cal-culated in the local density approximation (LDA) using aplane-wave basis with cutoff energy 400 eV. The 4s and4p semicore electrons of Mo were explicitly included in allcalculations. Unless otherwise stated the calculations havebeen performed for the experimental lattice constant of 3.16 A.One-shot G0W0 calculations were performed using the LDAwave functions and eigenvalues to obtain the G0W0@LDA

Γ K M Γk vector

-2

-1

0

1

2

3

4

Ene

rgy

(eV

)

LDAG

0W

0@LDA

FIG. 1. (Color online) Band structure of monolayer MoS2 cal-culated with LDA and G0W0@LDA using 45 × 45 k points and atruncated Coulomb interaction to avoid interaction between periodi-cally repeated layers. The valence-band tops have been aligned.

quasiparticle energies. A plane-wave cutoff of 50 eV and200 bands were used for the dielectric function, screenedinteraction, and GW self-energy. Convergence with respectto these parameters has been checked very carefully. Withthese values, band gaps were found to be converged withinaround 10 meV. The plasmon pole approximation of Godbyand Needs31 for the dielectric function was found to yield QPenergies within 0.1 eV of those obtained from full frequencydependence and was consequently used in all calculations.To avoid interaction between the periodically repeated MoS2

sheets, we have applied a truncated Coulomb interaction of theform vc(r) = (1/r)θ (Rc − z), following Refs. 32 and 33, in thecalculation of both the dielectric function, ε, and the screenedpotential, W = ε−1vc. For details on the implementation ofthe GW method in the GPAW code we refer to Ref. 34. Wenote that we have performed a numerical integration of theCoulomb interaction around each q point in the Brillouinzone [see Eq. (9) in Sec. III C] when evaluating the head ofthe screened potential, W00(q). This was found to be crucial forthe k-point convergence both when employing the truncatedand full Coulomb interaction.

B. Results

The band structure calculated using 45 × 45 k points andthe truncated Coulomb interaction is shown in Fig. 1. At theLDA level, we find a direct band gap at the K point of 1.77 eV,while the smallest indirect gap of 1.83 eV occurs from � to apoint along the �-K direction. In contrast, G0W0 predicts anindirect gap of 2.58 eV and a direct gap at K of 2.77 eV.

In Fig. 2 we show the convergence of both the direct and theindirect band gap with respect to the k-point grid for a fixedinterlayer separation of 23 A (see Fig. 3 for the definition ofL). It is clear that a very dense k-point grid is needed in orderto obtain well-converged results with the truncated Coulombinteraction. For 45 × 45 k points, band gaps are convergedwithin less than 0.1 eV, while this is already the case for15 × 15 k points with the bare Coulomb interaction. However,

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HOW DIELECTRIC SCREENING IN TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 88, 245309 (2013)

10 20 30 40 50

Number of k points

2

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

3.8

4

Ban

d ga

p (e

V)

1/r interactiontruncated interaction

FIG. 2. (Color online) Direct (full symbols) and indirect (opensymbols) G0W0 band gaps as function of the number of k points inone of the in-plane directions for a layer separation of L = 23 A.

the values obtained with the bare Coulomb interaction are toolow as a result of interlayer screening. The slow convergencewith respect to k points when the truncation is used will bediscussed in detail in Sec. III C.

We see that results do not converge independently with re-spect to the number of k points and the interlayer separation.35

In Fig. 4, we plot the L dependence of the direct band gapfor different k-point samplings with the bare interaction. Thek-point dependence becomes much stronger for large L. ForL → ∞, the values are expected to converge to the resultscalculated with the truncation (indicated by dotted lines). Theyseem to exhibit a linear 1/L behavior only for L > 50 A.Figure 5 shows all results and interpolated values in a contourplot as a function of 1/L and the number of k points. Theeffects of using more k points and increasing L are of differentsign and partially cancel each other. This is the reason whydifferent choices of the two parameters yield the same results.Especially, the band gaps calculated with 9 × 9 k points andL = 23 A and 15 × 15 k points and L = 43 A are the sameas with 45 × 45 k points and infinite L. This seems, however,coincidental, and we do not expect it to be the case for othersystems.

We note that all calculations have been performed witha single k point in the direction perpendicular to the layer.This is, however, insufficient for small interlayer distances.

FIG. 3. (Color online) Definition of the interlayer separation, L.

1020304060∞L (Å)

2.2

2.4

2.6

2.8

3

3.2

3.4

3.6

Dir

ect b

and

gap

(eV

)

(9×9×1) k points(15×15×1) k points(21×21×1) k points(33×33×1) k points(45×45×1) k points (truncated v

c)

FIG. 4. (Color online) Direct G0W0 band gap plotted as a functionof interlayer distance for different k-point samplings with the full 1/r

interaction. Dotted lines serve as a guide for the eye to extrapolatefor L → ∞. They were obtained by fitting all values for L > 30 A,including the results with the Coulomb truncation, to a quadraticfunction. Dashed horizontal lines indicate the calculated values withthe truncated Coulomb interaction.

For L = 13 A, we find an increase of the band gap of around0.2–0.3 eV when at least 3 k points are used, for example. ForL > 20 A or use of the truncation, this effect is negligible.

C. Comparison with previous work

In Table I we show our converged results obtained with thetruncated Coulomb interaction and 45 × 45 k points togetherwith previous G0W0 results from the literature. For eachreference we show the values used for the lattice constant,the interlayer separation, and the k-point sampling. It can beseen that all the previous calculations have used small layerseparations and no truncation method. As pointed out in the

FIG. 5. (Color online) Contour plot of the direct G0W0 band gapas a function of the inverse interlayer distance and number of k pointsin one of the in-plane directions with the full 1/r interaction. Contourlines are separated by 0.1 eV. Interpolation from splines was used.

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HUSER, OLSEN, AND THYGESEN PHYSICAL REVIEW B 88, 245309 (2013)

TABLE I. Calculated G0W0 band gaps obtained in the present work and compared with previous results from the literature. All ourcalculations have been performed using a truncated Coulomb interaction.

Egap (eV)

Reference Starting point a (A) Number of k points Layer separation (A) Direct Indirect

This work LDA 3.16 45 × 45 × 1 23 (truncated vc) 2.77 2.58This work LDA 3.19 45 × 45 × 1 23 (truncated vc) 2.65 2.57This work LDA 3.255 45 × 45 × 1 23 (truncated vc) 2.41 2.51Ref. 25 LDA 3.15 18 × 18 × 1 24 2.41 ∼2.40Ref. 4 PBE 3.16 12 × 12 × 1 19 ∼2.60 2.49Ref. 4 PBE 3.19 12 × 12 × 1 19 2.50 ∼2.55Ref. 4 PBE 3.255 12 × 12 × 1 19 2.19 2.19Ref. 21 LDA 3.16 8 × 8 × 2 19 2.96 –Ref. 22 PBE 3.18 12 × 12 × 1 20+1/L extrapolation 2.97 3.26Ref. 22 PBE 3.18 12 × 12 × 1 20 ∼2.60 ∼2.85Ref. 23 LDA 3.11 12 × 12 × 1 13 2.57 –Ref. 24 HSE 3.18 6 × 6 × 1 15 2.82 ∼3.00Ref. 36 PBE 3.19 15 × 15 × 1 15 2.66 –

preceding discussion, this gives a fast k-point convergence.A properly converged calculation, however, requires largerseparations and thereby more k points, but as a consequenceof a cancellation of errors a calculation with 19 A of layerseparation and 12 × 12 k points yields almost the same bandgaps as our converged result (within 0.15 eV). We are thus ledto conclude that the reasonable agreement between our resultsand previous ones is to a large extent fortuitous.

Furthermore, the effect of strain can have a large impact onthe MoS2 band gap. As demonstrated in Ref. 4, using 12 × 12k points and 19 A of layer separation, the G0W0 band gap forthe experimental lattice constant of 3.160 A is indirect. Witha lattice constant of 3.190 A, corresponding to 1% strain, thegap changes to be direct. The lowering of the direct band gapbecomes even more pronounced for larger lattice constants.As can be seen from the table, our converged results predictthe same trend, in particular the decrease of the direct gap asa function of strain, with our values for the direct gap beinggenerally 0.2 eV larger. We note that for 3.255 A the smallestindirect transition occurs from the � point at the valence bandto the K point at the conduction band. This is also in agreementwith Ref. 4. In the partially self-consistent GW0 calculationsof Ref. 26, the opposite trend was found, namely, a transitionfrom a direct to an indirect band gap for ∼5 % strain. However,a layer separation of only 12 A and less than 9 × 9 k points inthe in-plane directions were used in that study.

In Ref. 22, the band gap was determined by extrapolat-ing from L = 20 A to infinite layer separation, under theassumption that the gap scales linearly with the inversedistance between the layers. The obtained values for the directand indirect band gaps are ∼3.0 and ∼3.3 eV, respectively.This is consistent with our findings using the truncatedCoulomb interaction, the same lattice constant of 3.18 A,and the same (underconverged) k-point grid of 12 × 12 as inRef. 22.

From our studies, we conclude that the G0W0@LDAband gap of monolayer MoS2 is indirect with a value of2.6 eV, while the direct gap is 2.8 eV, when the experimentallattice constant of 3.16 A is used. The question of how well

the one-shot G0W0@LDA approximation describes the trueelectronic structure of this system remains open. Partiallyself-consistent GW0

4 and fully self-consistent GW 21 calcula-tions have been shown to consistently yield direct band gaps of2.75–2.80 eV for a layer separation of 19 A and a k-pointsampling of 12 × 12 × 1 and 8 × 8 × 2, respectively.

III. STATIC SCREENING

In this section we present a detailed investigation of the(static) dielectric properties of monolayer MoS2. This servesa dual purpose. First, it illustrates the origin of the slowconvergence of the GW results presented in the previoussection (and the BSE results presented in the next section).Second, it shows that the usual definition of the macroscopicdielectric constant of a periodic solid is not meaningful whenapplied to a 2D system represented in a periodic supercell. Wediscuss the difference between 2D and 3D screening, whichbecomes particularly pronounced in the q → 0 limit with largeconsequences for the calculation of optical excitations withstatic screening of the electron-hole interaction (see the nextsection).

A. 3D macroscopic dielectric constant

The microscopic dielectric function determines the relationbetween a weak external potential and the total potential in thematerial:

Vtot(r) =∫

dr′ ε−1(r,r′)Vext(r′). (1)

For a periodic system the dielectric function can be conve-niently expressed in plane waves:

ε−1(r,r′) =∑GG′

∑q

ei(G+q)rε−1GG′(q)e−i(G′+q)r′

, (2)

where G is a reciprocal lattice vector and q is a wave vector inthe 1. BZ. Within the random-phase approximation (RPA) wehave

εGG′(q,ω) = δGG′ − vc(q + G)χ0GG′(q,ω), (3)

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HOW DIELECTRIC SCREENING IN TWO-DIMENSIONAL . . . PHYSICAL REVIEW B 88, 245309 (2013)

where χ0 is the noninteracting density response function. Here,vc can be the Fourier representation of either the full or thetruncated Coulomb interaction. For the calculations in thissection we have used 50 eV cutoff for the reciprocal-latticevectors to account for local-field effects. The noninteractingresponse function, χ0, was constructed from LDA wavefunctions and energies including states up to 50 eV abovethe Fermi level. All calculations were performed with theprojector augmented wave method code GPAW. Details on theimplementation of the dielectric function in the GPAW code canbe found in Ref. 37.

It follows from Eq. (2) that the total potential resulting froma plane-wave external potential V0e

iq·r has the form

Vtot(r) = Vq(r)eiq·r, (4)

where Vq(r) is a lattice periodic function. We thus define themacroscopic dielectric constant as

1

εM (q)≡ 〈Vq〉�

V0= ε−1

00 (q), (5)

where 〈. . .〉� denotes a spatial average over a unit cell. Notethat in general εM (q,ω) = ε00(q,ω) because of local-fieldeffects.38,39

To explicitly demonstrate that Eq. (5) does not apply to low-dimensional materials, we have calculated the macroscopicdielectric constant as a function of the layer separation, L. Theresults are shown in Fig. 6 for different values of the in-planemomentum transfer q. We also show the dielectric constantcorresponding to polarization orthogonal to the layer. Clearly,the macroscopic dielectric constant approaches unity for all q

vectors in the limit of large interlayer separation. This occursbecause the total field is averaged over an increasingly largervacuum region.

Previously reported values for the macroscopic dielectricconstant of monolayer MoS2 lie in the range 4–8.21,24,28

In these calculations the MoS2 layers were separated by a10–20-A vacuum. As can be seen from ε‖(q = 0) in Fig. 6,

FIG. 6. (Color online) The 3D static macroscopic dielectricconstant 1/ε−1

00 (q) of monolayer MoS2 as a function of the interlayerseparation, L. ε‖ is the dielectric constant with polarization parallelto the monolayer, and ε⊥ is the dielectric constant for polarizationorthogonal to the layer.

this is consistent with our results. However, it should also beclear that numbers depend on the distance between layers andin fact are not meaningful.

B. 2D macroscopic dielectric constant

For a 2D material, the average of the total potential inthe definition of the macroscopic dielectric constant must beconfined to the region of the material. Since Eq. (4) stillholds for a 2D material when q is confined to the planeof the material, we average the in-plane coordinates (r‖)over the unit-cell area A and the out-of-plane coordinate (z)from z0 − d/2 to z0 + d/2, where z0 denotes the center ofthe material and d denotes its width. The 2D macroscopicdielectric constant then becomes

1

ε2DM (q‖)

≡ 〈Vq〉A,d

V0

= 2

d

∑G⊥

eiG⊥z0sin(G⊥d/2)

G⊥ε−1

G0 (q‖), (6)

where the sum is over all G with G‖ = 0. In this work we havetaken d = 6.15 A corresponding to the interlayer separationin bulk MoS2. We shall return to the problem of choosing d

below.The results for the static dielectric constant evaluated from

Eq. (6) using the bare Coulomb interaction are shown inFig. 7 for four different layer separations. The result forL = d = 6.15 A coincides with the 3D dielectric constant ofbulk MoS2 given by Eq. (5). The result obtained with thetruncated Coulomb interaction is shown in black; it representsthe case of infinite layer separation. Before discussing theresults, it is instructive to consider the potential arising from a

FIG. 7. (Color online) Static macroscopic dielectric constant fora single layer of MoS2 calculated along the �-K line. The calculationsare performed using Eq. (6) with the microscopic dielectric constant,ε−1

GG′ (q), evaluated from Eq. (3) with either the bare Coulombinteraction (dotted and dashed lines) or truncated Coulomb interaction(full black line). The gray area represents the result obtained whenthe averaging region perpendicular to the layer, d , is varied by ±10%.The dotted line corresponds to a layer separation of 6.15 A and thuscoincides with the dielectric constant of bulk MoS2. The curves havebeen interpolated from a 32 × 32 q-point mesh.

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HUSER, OLSEN, AND THYGESEN PHYSICAL REVIEW B 88, 245309 (2013)

2D charge-density fluctuation of the form

n(r) = n0eiq‖·r‖δ(z). (7)

The corresponding potential follows from Poisson’sequation:40

φ(r) = n0

q‖e−iq‖·r‖e−q‖|z|. (8)

It follows that the potential perpendicular to the layer fallsoff exponentially over a characteristic distance of 1/q‖. Thisexplains why in general ε2D

M (q‖) coincides with the isolatedlayer result for q‖ � 1/L.

The variation of ε2DM when the parameter d is changed by

±10% is indicated by the shaded region in Fig. 7. To the leftof the maximum, ε2D

M (q‖) is insensitive to d since the inducedpotential is more or less constant over the averaging region. Tothe right of the maximum, the variation in ε2D

M (q‖) follows the±10% variation in d. This is because for these wave vectorsthe induced potential has essentially vanished at the bordersof the averaging region. In general, increasing (decreasing)d will decrease (increase) ε2D

M (q‖) in the large wave-vectorregion.

Another characteristic feature of the potential in Eq. (8) isthe 1/q‖ scaling, which should be compared with the 1/q2

form of the Coulomb potential from a 3D charge oscillation.Since the noninteracting response function, χ0

00(q), scales as∼q2 for q → 0 for both 2D and 3D systems, it followsfrom Eq. (3) that ε2D

M (0) = 1, while this is in general notthe case in three dimensions. In our calculations, the effectof interlayer interactions is eliminated by using a truncatedCoulomb interaction of the form vc(r) = (1/r)θ (Rc − z). Forqz = 0 and in the limit of small q‖, the Fourier representationof the truncated Coulomb interaction becomes v2D(q) = 4πRc

|q| ;i.e., it scales as 1/q as the potential from the 2D charge-densitywave, ensuring the correct limit ε2D

M (0) = 1.Finally, we note that previous studies41,42 have employed a

strict 2D model for the dielectric function in the small q limit ofthe form ε(q‖) = 1 + αq‖. This form is convenient, as it leadsto an analytical expression for the 2D screened interaction.27

Our definition differs by being a 3D (or quasi-2D) quantityvalid for general q‖.

C. Screened interaction

In Fig. 8 we show ε−100 as a function of q‖ evaluated with and

without the truncated Coulomb interaction. For small q, thetwo curves differ significantly due to the long-range nature ofthe induced potential Eq. (8). At large q (∼K/2), the inducedpotential decays within the cutoff range for the truncatedCoulomb interaction, and therefore no difference can be seenbetween the two methods. We emphasize that neither of thedielectric constants shown in the figure can be interpreted asa dielectric constant of monolayer MoS2, since they give theaverage potential over the supercell and not over the MoS2

layer. In particular, their value will be highly dependent on thesize of the unit cell (in the limit of infinite layer separation,both will equal 1 for all q). Nevertheless, this quantity is acrucial ingredient of both the GW self-energy and the BSEkernel, as it provides the screening of the divergent term of theCoulomb interaction.

Γ K/4wave vector q

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

∋-1 00(q

, ω=

0)

1/r interactiontruncated interaction

FIG. 8. (Color online) The 3D static inverse dielectric constantε−1

00 (q) of monolayer MoS2 calculated in the RPA for differentvalues of in-plane momentum transfer q along the �-K direction.The separation between layers is L = 20 A. Note that neither of thequantities can be interpreted as the macroscopic dielectric constantof the monolayer (this quantity is the black curve in Fig. 7).

For q = 0 the Coulomb kernel diverges and we approximateW (q = 0) by the integral

W00(q = 0) = 1

��

∫��

dq vc(q)ε−100 (q)

≈ 1

��

ε−100 (q = 0)

∫��

dq vc(q), (9)

where �BZ is the Brillouin-zone volume and �� is a smallvolume containing q = 0. In isotropic systems, ε−1

00 (q) isusually constant in the vicinity of q = 0 and the approximationworks well. However, when ε−1 is evaluated with the truncatedCoulomb interaction, ε−1

00 acquires much more structure forsmall q as can be seen from Fig. 8. Thus, for coarse k-pointsamplings we will underestimate the �-point screening sincewe simply use ε−1

00 (q = 0) = 1.The linear behavior of the screened interaction for small q

suggests that a better approximation for W00(q = 0) would be

W00(q = 0) = 1

��

∫��

dq vc(q)[1 + q · ∇qε

−100 (q)

∣∣q=0

]. (10)

Since the dielectric matrix in RPA is εGG′(q) = 1 −vc(q)χ0

GG′(q), we can derive an analytic expression for the first-order Taylor expansion in q and its inverse. These quantitiescan be evaluated with vanishing additional cost, but we willleave the assessment of this approximation to future work.

IV. OPTICAL ABSORPTION SPECTRUM

In this section we present many-body calculations of theoptical absorption spectrum of monolayer MoS2 by solving theBethe-Salpeter equation (BSE) under the standard assumptionof static screening of the electron-hole interaction. As for theGW band gap, we find that the use of a truncated Coulombinteraction is essential to avoid interlayer screening and obtainwell-converged exciton binding energies. Furthermore, thevery strong q dependence of the 2D static dielectric function

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around q = 0 leads to very slow k-point convergence for theexciton binding energy.

In order to obtain an accurate absorption spectrum includingexcitonic effects we calculate the response function from theBSE. Using the standard assumption of a static dielectricscreening of the electron-hole (e-h) interaction, the BSE43 canbe recast as an effective two-particle Hamiltonian,44 which isdiagonalized in a basis of electron-hole pairs. In this way theexcitonic eigenstates can be expressed as a linear combinationof single-particle transitions:

|λ〉 =∑vck

Aλvck|vck〉, (11)

where v, c, and k denote valence bands, conduction bands,and Brillouin-zone wave vectors, respectively. The absorp-tion spectrum is proportional to the imaginary part of themacroscopic dielectric function, which in the Tamm-Dancoffapproximation can be written

ε2(ω) = 2π limq→0

vc(q)∑

λ

δ(ω − Eλ)

×∣∣∣∣∑

vck

Aλvck〈vk − q|e−iq·r|ck〉

∣∣∣∣2

, (12)

where Eλ are the eigenvalues associated with |λ〉.In all calculations we have included a single valence band

and a single conduction band in the BSE Hamiltonian. We havetested that the first excitonic peak is completely unaffected ifwe instead include six valence bands and four conductionbands. This is also expected since the highest (lowest) valence(conduction) band is well isolated from the remaining bands atK where the exciton is centered (see Fig. 1). For the screeningwe have included 65 bands in the evaluation of the responsefunction, which is sufficient for converged results. Increasingthe number of bands to 300 affects the position of the firstexciton by less than 10 meV. The plane-wave cutoff for theresponse function (local-field effects) was set to 50 eV, and wechecked that the excitonic binding energy changed by less than10 meV when increasing the cutoff to 200 eV. The dependenceon k-point sampling and interlayer separation will be examinedbelow. Details on the implementation of the BSE method inthe GPAW code can be found in Ref. 5.

A. Convergence tests

In the lower panel of Fig. 9, we show the exciton bindingenergy as a function of interlayer separation calculated fordifferent k-point samplings using the bare Coulomb interactionand the truncated Coulomb interaction. With the bare Coulombinteraction, the obtained results are far from convergence, evenfor L = 50 A. The dependencies on the layer separation andnumber of k points are very similar as for the quasiparticle gapdiscussed in Sec. II B, even on a quantitative level. Therefore,the optical gap, which is given by the difference of the QPgap and the exciton binding energy, is almost independent ofL and whether or not the truncation method is used, as shownin the upper panel. This is consistent with the observations inRef. 22.

The convergence of the binding energy with respect tothe k-point sampling is plotted in Fig. 10 for an interlayer

1020304060∞L (Å)

0.2

0.4

0.6

0.8

1

Bin

ding

ene

rgy

(eV

)

(15×15×1) k points(30×30×1) k pointstruncated interaction1.8

2

2.2

Opt

ical

gap

(eV

)

FIG. 9. (Color online) Optical gap and binding energy of thelowest exciton in monolayer MoS2 as a function of interlayerseparation calculated from the BSE and the G0W0 quasiparticle gap.Results with the full 1/r Coulomb interaction (full lines) and thetruncated interaction (dashed lines) are shown. Dotted lines give anestimation for extrapolation to infinite L.

separation of 20 A. The truncated Coulomb kernel gives amuch slower convergence with respect to the number of k

points than the bare Coulomb interaction. However, it shouldbe clear from Fig. 9 that the binding energy obtained with thebare Coulomb interaction converges to a value which is highlydependent on the interlayer separation. In Ref. 25, convergencewas found with 18 × 18 k points, but for a layer separation ofonly 24 A. The obtained exciton binding energy was around0.2 eV. According to our results, this is much too weak due tointerlayer screening.

The slow k-point convergence observed when using thetruncated Coulomb interaction is related to the q dependenceof the screening in two-dimensional systems. As demonstratedby Eq. (9) and Fig. 8 (purple curve), a too low k-point samplingleads to an underestimation of the screening in the vicinity

0 10 20 30 40 50 60

Number of k points

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Bin

ding

ene

rgy

(eV

)

1/r interactiontruncated interaction

FIG. 10. (Color online) Binding energy of the lowest exciton inmonolayer MoS2 as a function of k-point sampling for a supercellwith a layer separation of L = 20 A.

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FIG. 11. (Color online) Absorption spectrum of single layerMoS2 calculated with the RPA and BSE using the G0W0 quasiparticleband structure. The calculation has been performed with a truncatedCoulomb interaction to avoid interactions between repeated layersand with a 45 × 45 k-point grid.

of q = 0 and thus an overestimation of the exciton bindingenergy.

We note that the slight increase in the binding energy withk points obtained when using the full Coulomb interactionis not a result of decreased screening. Indeed the dielectricconstant for this case is well converged already for a k-pointsampling of 15 × 15 (see the slow wave vector dependence ofthe blue curve in Fig. 8). Instead, it is simply a result of theincreased number of electron-hole basis states used to describethe exciton which lowers the BSE eigenvalue slightly (andthus increases the exciton binding energy). This effect is alsopresent when using the truncated Coulomb interaction, but inthis case the k-point variation in binding energy is dominatedby the effect of increasing screening as discussed above (notethat the screening effect is missed if W is calculated on a coarsegrid and extrapolated to a fine k-point grid before solving theBSE).

B. Results

From the convergence tests described above we concludethat the BSE calculations are (nearly) converged if we usea truncated Coulomb interaction and a 45 × 45 k-point sam-pling. With these settings we have calculated the BSE spectrumon top of a G0W0 quasiparticle band structure obtained with thesame parameters. The resulting absorption spectrum is shownin Fig. 11. We also show an RPA calculation, i.e., neglectingelectron-hole interactions in the BSE, performed on top of thesame G0W0 band structure for comparison. With electron-holeinteraction included, we obtain an exciton binding energy of∼0.6 eV, whereas RPA does not show an exciton peak andsimply gives an absorption edge at the band gap.

Experimentally, the absorption spectrum of single layerMoS2 exhibits a spin-orbit split peak around 1.9 eV.7 Sincewe have not included spin-orbit coupling in our calculations,the spectrum in Fig. 11 only shows a single peak at lowenergies. However, it has previously been shown4,24 that the

spin-orbit coupling does not have a large effect on the excitonbinding energy and only results in a split excitonic peak. Themain peak in the BSE@G0W0 spectrum is situated at 2.2 eV,which is 0.3 eV higher than the experimental value. At presentwe cannot say if this is due to an insufficient descriptionof the quasiparticle gap within G0W0 or underestimation ofthe exciton binding energy in BSE with a static electron-holeinteraction.

From the above discussion it should be clear that it isextremely challenging to converge the exciton binding energywith respect to interlayer separation and k points. In general,the optical gap is much easier to converge with respect tointerlayer separation than either the quasiparticle gap or theexciton binding energy.22,45 Nevertheless, for many physicalapplications it is of importance to obtain accurate values forboth the quasiparticle gap and the exciton binding energyseparately. In Ref. 22 the exciton binding energy was obtainedby 1/L extrapolation of the quasiparticle gap calculated ina range of interlayer separations between 10 and 20 A andassuming the same dependence for the exciton binding energy.Our results indicate that one should be cautious with suchextrapolations. This is because the screening at different q

points has a very different dependence on interlayer separation,which results in different convergence behavior at differentk-point samplings (see Fig. 9, full lines). The extrapolationprocedure may therefore not give reliable results, since higherk-point samplings are required at larger interlayer separation.We are aware that the convergence issues may depend a loton the implementation of the BSE method. However, wehave previously performed the same calculations with theYAMBO46 code, which produced very similar convergencebehavior for quasiparticle gap and exciton binding energy(also using the truncated Coulomb cutoff and 45 × 45 k-pointsampling).

V. CONCLUSIONS

We have presented a careful investigation of the dielectricproperties, band gap, and excitonic states in a two-dimensionalsemiconductor exemplified by monolayer MoS2. We havedemonstrated that the “standard” macroscopic dielectric con-stant used for solids is not applicable (meaningless) tosupercells describing the 2D material as an infinite array ofparallel sheets and therefore replaced it by a 2D version inwhich the induced field is averaged over the extent of thematerial rather than over the entire supercell. We showed thatthe effect of interlayer screening leads to underestimation ofthe band gap and exciton binding energy by up to more than0.5 eV for layer separations <30 A. The reason for this isthat interlayer screening affects ε(q) for q < 1/L, where L

is the distance between layers in the supercell. Since it is thesmall q limit of ε(q) that is most important for the screenedinteraction W (q) = ε−1(q)/q2, the effect cannot be neglected.Here we have circumvented the problem by using a truncatedCoulomb interaction that explicitly cuts off the interactionbetween neighboring layers.

The properly defined 2D dielectric function ε2DM (q) has a

very sharp wave-vector dependence for small q and satisfiesε2DM (0) = 1 in general. This has the consequence that quasi-

particle and optical excitations obtained from the GW and

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Bethe-Salpeter equation, respectively, require much denserk-point grids than experience from 3D systems would suggest.For MoS2 we find that a precision of 0.2 eV requires k-pointgrids of at least 30 × 30. Interestingly, the effects of interlayerscreening and too small k-point grids have opposite effects onthe band gap and exciton energies, leading to fortuitous errorcancellation. Our calculations applying the truncated Coulombinteraction and 45 × 45 k points give G0W0@LDA gaps of2.77 eV (direct) and 2.58 eV (indirect) and a binding energyof the lowest exciton of 0.55 eV. This places the lowest excitonat ∼2.2 eV, which is 0.3 eV higher than the experimental result.

This difference may be due to the G0W0@LDA approximationor the use of static screening in the BSE.

ACKNOWLEDGMENTS

K.S.T. acknowledges support from the Danish Council forIndependent Research Sapere Aude Program through GrantNo. 11-1051390. The Center for Nanostructured Grapheneis sponsored by the Danish National Research Foundation,Project No. DNRF58.

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